I grew up riding on horse trails--we had a set of great trails within riding distance from my house. I *never* picked up after the horse. Ever. Not once.
This is how I look at it---The trails are set aside for horse back riders and hikers March through November and cross country skiers through the winter. Both horse back riding and cross country skiing are hobbies that take a toll on the trail. Horses leave manure and the skiers leave lots of ruts in the snow that harden.
The things is, you can hike anywhere on any trail in any city, county, state, or national park, but you cannot horseback ride (or ski) in most of those locations. Therefore, if the hikers (who can go anywhere they want pretty much) choose to use trails also frequented by horses, then they need to accept that there is going to be poo, just as if they are going to share the trails with skiers in the winter, they need to be prepared to deal with ruts in the snow (which are *far* more dangerous than poo, BTW, I know this from experience). If poo or ruts in the snow are a problem, there is always the option of a number of other hiking locations.
As Dekka pointed out, repeated mounting from the ground without the aide of a block can be physically hard on a horse (some horses poop smaller amounts a number of times during the ride), some riders have physical difficulties getting on their horses from the ground (therefore by requiring dismounting, you are essentially closing off all trail riding to riders with physical challenges), and there is a lot of potential for the trail traffic to be backed up while a rider tries to deal with an uncooperative horse--resulting in the very least in annoyed park users, and at worst in a horse or person getting hurt. There is a *huge* difference between a 60 lb dog pulling on the leash and a 1200 lb horse 'pulling on the leash."
Tons of potential problems for what? So people don't have to move 12 inches to the side to avoid some horse poo?